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Combinatorics and Graph Theory — One Question at a Time

Combinatorics and Graph Theory — One Question at a Time

Practise 200 Combinatorics and Graph Theory multiple-choice questions covering counting methods, permutations, combinations, graphs, trees, shortest paths and asymptotic notation. Work through one question at a time, check the correct answer with an explanation, and select Next Question to continue. This is self-paced practice without automatic scoring.

Before you practise: Read Combinatorics and Graph Theory study material and examples. You can also browse all 1,576 interactive MCQs.

Practice Set 1 · Question 1 of 200

Question 1. How many ways can 3 distinct books be arranged on one shelf?

  1. 6
  2. 2
  3. 18
  4. 9
Show Answer and Explanation

Correct answer: A. 6

There are 3! permutations of 3 distinct objects.

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Practice Set 1 · Question 2 of 200

Question 2. A student chooses one of 3 shirts and one of 4 pairs of trousers. How many outfits are possible?

  1. 8
  2. 12
  3. 7
  4. 11
Show Answer and Explanation

Correct answer: B. 12

By the product rule: 3×4=12.

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Practice Set 1 · Question 3 of 200

Question 3. A menu offers 3 desserts or 5 drinks as a single choice (disjoint options). How many choices?

  1. 8
  2. 15
  3. 7
  4. 16
Show Answer and Explanation

Correct answer: A. 8

By the addition rule: 3+5=8.

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Practice Set 1 · Question 4 of 200

Question 4. How many ways can 4 distinct books be arranged on one shelf?

  1. 6
  2. 96
  3. 16
  4. 24
Show Answer and Explanation

Correct answer: D. 24

There are 4! permutations of 4 distinct objects.

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Practice Set 1 · Question 5 of 200

Question 5. A student chooses one of 4 shirts and one of 5 pairs of trousers. How many outfits are possible?

  1. 20
  2. 9
  3. 19
  4. 10
Show Answer and Explanation

Correct answer: A. 20

By the product rule: 4×5=20.

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Practice Set 1 · Question 6 of 200

Question 6. A menu offers 4 desserts or 6 drinks as a single choice (disjoint options). How many choices?

  1. 24
  2. 9
  3. 25
  4. 10
Show Answer and Explanation

Correct answer: D. 10

By the addition rule: 4+6=10.

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Practice Set 1 · Question 7 of 200

Question 7. How many ways can 5 distinct books be arranged on one shelf?

  1. 600
  2. 25
  3. 120
  4. 24
Show Answer and Explanation

Correct answer: C. 120

There are 5! permutations of 5 distinct objects.

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Practice Set 1 · Question 8 of 200

Question 8. A student chooses one of 5 shirts and one of 6 pairs of trousers. How many outfits are possible?

  1. 11
  2. 29
  3. 12
  4. 30
Show Answer and Explanation

Correct answer: D. 30

By the product rule: 5×6=30.

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Practice Set 1 · Question 9 of 200

Question 9. A menu offers 5 desserts or 7 drinks as a single choice (disjoint options). How many choices?

  1. 11
  2. 36
  3. 12
  4. 35
Show Answer and Explanation

Correct answer: C. 12

By the addition rule: 5+7=12.

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Practice Set 1 · Question 10 of 200

Question 10. How many ways can 6 distinct books be arranged on one shelf?

  1. 36
  2. 720
  3. 120
  4. 4320
Show Answer and Explanation

Correct answer: B. 720

There are 6! permutations of 6 distinct objects.

Continue to Practice Set 2

Practice Set 2 · Question 11 of 200

Question 11. A student chooses one of 6 shirts and one of 7 pairs of trousers. How many outfits are possible?

  1. 41
  2. 14
  3. 42
  4. 13
Show Answer and Explanation

Correct answer: C. 42

By the product rule: 6×7=42.

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Practice Set 2 · Question 12 of 200

Question 12. A menu offers 6 desserts or 8 drinks as a single choice (disjoint options). How many choices?

  1. 49
  2. 14
  3. 48
  4. 13
Show Answer and Explanation

Correct answer: B. 14

By the addition rule: 6+8=14.

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Practice Set 2 · Question 13 of 200

Question 13. How many ways can 7 distinct books be arranged on one shelf?

  1. 5040
  2. 720
  3. 35280
  4. 49
Show Answer and Explanation

Correct answer: A. 5040

There are 7! permutations of 7 distinct objects.

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Practice Set 2 · Question 14 of 200

Question 14. A student chooses one of 7 shirts and one of 8 pairs of trousers. How many outfits are possible?

  1. 16
  2. 56
  3. 15
  4. 55
Show Answer and Explanation

Correct answer: B. 56

By the product rule: 7×8=56.

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Practice Set 2 · Question 15 of 200

Question 15. A menu offers 7 desserts or 9 drinks as a single choice (disjoint options). How many choices?

  1. 16
  2. 63
  3. 15
  4. 64
Show Answer and Explanation

Correct answer: A. 16

By the addition rule: 7+9=16.

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Practice Set 2 · Question 16 of 200

Question 16. How many ways can 8 distinct books be arranged on one shelf?

  1. 5040
  2. 322560
  3. 64
  4. 40320
Show Answer and Explanation

Correct answer: D. 40320

There are 8! permutations of 8 distinct objects.

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Practice Set 2 · Question 17 of 200

Question 17. A student chooses one of 8 shirts and one of 9 pairs of trousers. How many outfits are possible?

  1. 72
  2. 17
  3. 71
  4. 18
Show Answer and Explanation

Correct answer: A. 72

By the product rule: 8×9=72.

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Practice Set 2 · Question 18 of 200

Question 18. A menu offers 8 desserts or 10 drinks as a single choice (disjoint options). How many choices?

  1. 80
  2. 17
  3. 81
  4. 18
Show Answer and Explanation

Correct answer: D. 18

By the addition rule: 8+10=18.

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Practice Set 2 · Question 19 of 200

Question 19. How many ways can 9 distinct books be arranged on one shelf?

  1. 3265920
  2. 81
  3. 362880
  4. 40320
Show Answer and Explanation

Correct answer: C. 362880

There are 9! permutations of 9 distinct objects.

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Practice Set 2 · Question 20 of 200

Question 20. A student chooses one of 9 shirts and one of 10 pairs of trousers. How many outfits are possible?

  1. 19
  2. 89
  3. 20
  4. 90
Show Answer and Explanation

Correct answer: D. 90

By the product rule: 9×10=90.

Continue to Practice Set 3

Practice Set 3 · Question 21 of 200

Question 21. A menu offers 9 desserts or 11 drinks as a single choice (disjoint options). How many choices?

  1. 19
  2. 100
  3. 20
  4. 99
Show Answer and Explanation

Correct answer: C. 20

By the addition rule: 9+11=20.

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Practice Set 3 · Question 22 of 200

Question 22. How many ways can 10 distinct books be arranged on one shelf?

  1. 100
  2. 3628800
  3. 362880
  4. 36288000
Show Answer and Explanation

Correct answer: B. 3628800

There are 10! permutations of 10 distinct objects.

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Practice Set 3 · Question 23 of 200

Question 23. A student chooses one of 10 shirts and one of 11 pairs of trousers. How many outfits are possible?

  1. 109
  2. 22
  3. 110
  4. 21
Show Answer and Explanation

Correct answer: C. 110

By the product rule: 10×11=110.

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Practice Set 3 · Question 24 of 200

Question 24. A menu offers 10 desserts or 12 drinks as a single choice (disjoint options). How many choices?

  1. 121
  2. 22
  3. 120
  4. 21
Show Answer and Explanation

Correct answer: B. 22

By the addition rule: 10+12=22.

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Practice Set 3 · Question 25 of 200

Question 25. How many ordered selections of 2 objects can be made from 6 distinct objects without replacement?

  1. 12
  2. 15
  3. 36
  4. 30
Show Answer and Explanation

Correct answer: D. 30

P(6,2) = 6!/(6−2)! = 30.

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Practice Set 3 · Question 26 of 200

Question 26. How many ordered selections of 3 objects can be made from 7 distinct objects without replacement?

  1. 70
  2. 343
  3. 210
  4. 21
Show Answer and Explanation

Correct answer: C. 210

P(7,3) = 7!/(7−3)! = 210.

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Practice Set 3 · Question 27 of 200

Question 27. How many ordered selections of 4 objects can be made from 8 distinct objects without replacement?

  1. 4096
  2. 1680
  3. 32
  4. 420
Show Answer and Explanation

Correct answer: B. 1680

P(8,4) = 8!/(8−4)! = 1680.

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Practice Set 3 · Question 28 of 200

Question 28. How many ordered selections of 2 objects can be made from 9 distinct objects without replacement?

  1. 72
  2. 18
  3. 36
  4. 81
Show Answer and Explanation

Correct answer: A. 72

P(9,2) = 9!/(9−2)! = 72.

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Practice Set 3 · Question 29 of 200

Question 29. How many ordered selections of 3 objects can be made from 10 distinct objects without replacement?

  1. 30
  2. 240
  3. 1000
  4. 720
Show Answer and Explanation

Correct answer: D. 720

P(10,3) = 10!/(10−3)! = 720.

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Practice Set 3 · Question 30 of 200

Question 30. How many ordered selections of 4 objects can be made from 11 distinct objects without replacement?

  1. 1980
  2. 14641
  3. 7920
  4. 44
Show Answer and Explanation

Correct answer: C. 7920

P(11,4) = 11!/(11−4)! = 7920.

Continue to Practice Set 4

Practice Set 4 · Question 31 of 200

Question 31. How many ordered selections of 2 objects can be made from 12 distinct objects without replacement?

  1. 144
  2. 132
  3. 24
  4. 66
Show Answer and Explanation

Correct answer: B. 132

P(12,2) = 12!/(12−2)! = 132.

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Practice Set 4 · Question 32 of 200

Question 32. How many ordered selections of 3 objects can be made from 13 distinct objects without replacement?

  1. 1716
  2. 39
  3. 572
  4. 2197
Show Answer and Explanation

Correct answer: A. 1716

P(13,3) = 13!/(13−3)! = 1716.

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Practice Set 4 · Question 33 of 200

Question 33. How many 2-element subsets can be selected from 7 distinct elements?

  1. 14
  2. 49
  3. 21
  4. 42
Show Answer and Explanation

Correct answer: C. 21

C(7,2) = 7!/[2!(7−2)!] = 21.

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Practice Set 4 · Question 34 of 200

Question 34. How many 3-element subsets can be selected from 8 distinct elements?

  1. 512
  2. 56
  3. 168
  4. 24
Show Answer and Explanation

Correct answer: B. 56

C(8,3) = 8!/[3!(8−3)!] = 56.

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Practice Set 4 · Question 35 of 200

Question 35. How many 4-element subsets can be selected from 9 distinct elements?

  1. 126
  2. 504
  3. 36
  4. 6561
Show Answer and Explanation

Correct answer: A. 126

C(9,4) = 9!/[4!(9−4)!] = 126.

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Practice Set 4 · Question 36 of 200

Question 36. How many 2-element subsets can be selected from 10 distinct elements?

  1. 90
  2. 20
  3. 100
  4. 45
Show Answer and Explanation

Correct answer: D. 45

C(10,2) = 10!/[2!(10−2)!] = 45.

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Practice Set 4 · Question 37 of 200

Question 37. How many 3-element subsets can be selected from 11 distinct elements?

  1. 33
  2. 1331
  3. 165
  4. 495
Show Answer and Explanation

Correct answer: C. 165

C(11,3) = 11!/[3!(11−3)!] = 165.

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Practice Set 4 · Question 38 of 200

Question 38. How many 4-element subsets can be selected from 12 distinct elements?

  1. 20736
  2. 495
  3. 1980
  4. 48
Show Answer and Explanation

Correct answer: B. 495

C(12,4) = 12!/[4!(12−4)!] = 495.

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Practice Set 4 · Question 39 of 200

Question 39. How many 2-element subsets can be selected from 13 distinct elements?

  1. 78
  2. 156
  3. 26
  4. 169
Show Answer and Explanation

Correct answer: A. 78

C(13,2) = 13!/[2!(13−2)!] = 78.

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Practice Set 4 · Question 40 of 200

Question 40. How many 3-element subsets can be selected from 14 distinct elements?

  1. 1092
  2. 42
  3. 2744
  4. 364
Show Answer and Explanation

Correct answer: D. 364

C(14,3) = 14!/[3!(14−3)!] = 364.

Continue to Practice Set 5

Practice Set 5 · Question 41 of 200

Question 41. How many length-2 sequences can be formed from 3 symbols if repetition is allowed?

  1. 6
  2. 8
  3. 3
  4. 9
Show Answer and Explanation

Correct answer: D. 9

Each of the 2 positions has 3 choices, giving 3^2=9.

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Practice Set 5 · Question 42 of 200

Question 42. How many length-3 sequences can be formed from 4 symbols if repetition is allowed?

  1. 63
  2. 16
  3. 64
  4. 12
Show Answer and Explanation

Correct answer: C. 64

Each of the 3 positions has 4 choices, giving 4^3=64.

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Practice Set 5 · Question 43 of 200

Question 43. How many length-4 sequences can be formed from 5 symbols if repetition is allowed?

  1. 125
  2. 625
  3. 20
  4. 624
Show Answer and Explanation

Correct answer: B. 625

Each of the 4 positions has 5 choices, giving 5^4=625.

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Practice Set 5 · Question 44 of 200

Question 44. How many length-2 sequences can be formed from 6 symbols if repetition is allowed?

  1. 36
  2. 12
  3. 35
  4. 6
Show Answer and Explanation

Correct answer: A. 36

Each of the 2 positions has 6 choices, giving 6^2=36.

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Practice Set 5 · Question 45 of 200

Question 45. How many length-3 sequences can be formed from 7 symbols if repetition is allowed?

  1. 21
  2. 342
  3. 49
  4. 343
Show Answer and Explanation

Correct answer: D. 343

Each of the 3 positions has 7 choices, giving 7^3=343.

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Practice Set 5 · Question 46 of 200

Question 46. How many length-4 sequences can be formed from 8 symbols if repetition is allowed?

  1. 4095
  2. 512
  3. 4096
  4. 32
Show Answer and Explanation

Correct answer: C. 4096

Each of the 4 positions has 8 choices, giving 8^4=4096.

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Practice Set 5 · Question 47 of 200

Question 47. How many length-2 sequences can be formed from 9 symbols if repetition is allowed?

  1. 9
  2. 81
  3. 18
  4. 80
Show Answer and Explanation

Correct answer: B. 81

Each of the 2 positions has 9 choices, giving 9^2=81.

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Practice Set 5 · Question 48 of 200

Question 48. How many length-3 sequences can be formed from 10 symbols if repetition is allowed?

  1. 1000
  2. 30
  3. 999
  4. 100
Show Answer and Explanation

Correct answer: A. 1000

Each of the 3 positions has 10 choices, giving 10^3=1000.

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Practice Set 5 · Question 49 of 200

Question 49. If 4 objects are placed into 3 boxes, at least how many objects must share a box?

  1. 3
  2. 4
  3. 2
  4. 1
Show Answer and Explanation

Correct answer: C. 2

Since 4>3, some box contains at least two objects.

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Practice Set 5 · Question 50 of 200

Question 50. If 5 objects are distributed among 2 boxes, what minimum occupancy is guaranteed in some box?

  1. 4
  2. 3
  3. 1
  4. 2
Show Answer and Explanation

Correct answer: B. 3

Ceiling(5/2) = 3, so some box has at least three objects.

Continue to Practice Set 6

Practice Set 6 · Question 51 of 200

Question 51. If 5 objects are placed into 4 boxes, at least how many objects must share a box?

  1. 4
  2. 2
  3. 1
  4. 3
Show Answer and Explanation

Correct answer: B. 2

Since 5>4, some box contains at least two objects.

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Practice Set 6 · Question 52 of 200

Question 52. If 7 objects are distributed among 3 boxes, what minimum occupancy is guaranteed in some box?

  1. 3
  2. 1
  3. 2
  4. 4
Show Answer and Explanation

Correct answer: A. 3

Ceiling(7/3) = 3, so some box has at least three objects.

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Practice Set 6 · Question 53 of 200

Question 53. If 6 objects are placed into 5 boxes, at least how many objects must share a box?

  1. 2
  2. 1
  3. 3
  4. 5
Show Answer and Explanation

Correct answer: A. 2

Since 6>5, some box contains at least two objects.

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Practice Set 6 · Question 54 of 200

Question 54. If 9 objects are distributed among 4 boxes, what minimum occupancy is guaranteed in some box?

  1. 1
  2. 2
  3. 4
  4. 3
Show Answer and Explanation

Correct answer: D. 3

Ceiling(9/4) = 3, so some box has at least three objects.

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Practice Set 6 · Question 55 of 200

Question 55. If 7 objects are placed into 6 boxes, at least how many objects must share a box?

  1. 1
  2. 3
  3. 6
  4. 2
Show Answer and Explanation

Correct answer: D. 2

Since 7>6, some box contains at least two objects.

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Practice Set 6 · Question 56 of 200

Question 56. If 11 objects are distributed among 5 boxes, what minimum occupancy is guaranteed in some box?

  1. 2
  2. 4
  3. 3
  4. 1
Show Answer and Explanation

Correct answer: C. 3

Ceiling(11/5) = 3, so some box has at least three objects.

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Practice Set 6 · Question 57 of 200

Question 57. If 8 objects are placed into 7 boxes, at least how many objects must share a box?

  1. 3
  2. 7
  3. 2
  4. 1
Show Answer and Explanation

Correct answer: C. 2

Since 8>7, some box contains at least two objects.

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Practice Set 6 · Question 58 of 200

Question 58. If 13 objects are distributed among 6 boxes, what minimum occupancy is guaranteed in some box?

  1. 4
  2. 3
  3. 1
  4. 2
Show Answer and Explanation

Correct answer: B. 3

Ceiling(13/6) = 3, so some box has at least three objects.

Next Question →

Practice Set 6 · Question 59 of 200

Question 59. If 9 objects are placed into 8 boxes, at least how many objects must share a box?

  1. 8
  2. 2
  3. 1
  4. 3
Show Answer and Explanation

Correct answer: B. 2

Since 9>8, some box contains at least two objects.

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Practice Set 6 · Question 60 of 200

Question 60. If 15 objects are distributed among 7 boxes, what minimum occupancy is guaranteed in some box?

  1. 3
  2. 1
  3. 2
  4. 4
Show Answer and Explanation

Correct answer: A. 3

Ceiling(15/7) = 3, so some box has at least three objects.

Continue to Practice Set 7

Practice Set 7 · Question 61 of 200

Question 61. If 10 objects are placed into 9 boxes, at least how many objects must share a box?

  1. 2
  2. 1
  3. 3
  4. 9
Show Answer and Explanation

Correct answer: A. 2

Since 10>9, some box contains at least two objects.

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Practice Set 7 · Question 62 of 200

Question 62. If 17 objects are distributed among 8 boxes, what minimum occupancy is guaranteed in some box?

  1. 1
  2. 2
  3. 4
  4. 3
Show Answer and Explanation

Correct answer: D. 3

Ceiling(17/8) = 3, so some box has at least three objects.

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Practice Set 7 · Question 63 of 200

Question 63. If 11 objects are placed into 10 boxes, at least how many objects must share a box?

  1. 1
  2. 3
  3. 10
  4. 2
Show Answer and Explanation

Correct answer: D. 2

Since 11>10, some box contains at least two objects.

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Practice Set 7 · Question 64 of 200

Question 64. If 19 objects are distributed among 9 boxes, what minimum occupancy is guaranteed in some box?

  1. 2
  2. 4
  3. 3
  4. 1
Show Answer and Explanation

Correct answer: C. 3

Ceiling(19/9) = 3, so some box has at least three objects.

Next Question →

Practice Set 7 · Question 65 of 200

Question 65. How many edges are in the complete graph K4?

  1. 6
  2. 4
  3. 3
  4. 12
Show Answer and Explanation

Correct answer: A. 6

Every unordered pair of vertices forms an edge: 4(4−1)/2.

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Practice Set 7 · Question 66 of 200

Question 66. How many edges does a simple cycle graph C4 have?

  1. 3
  2. 5
  3. 6
  4. 4
Show Answer and Explanation

Correct answer: D. 4

A cycle has one edge for each of its 4 vertices.

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Practice Set 7 · Question 67 of 200

Question 67. How many edges are in the path graph P4 with 4 vertices?

  1. 5
  2. 8
  3. 3
  4. 4
Show Answer and Explanation

Correct answer: C. 3

A path through 4 vertices uses 4−1 edges.

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Practice Set 7 · Question 68 of 200

Question 68. How many edges are in the complete graph K5?

  1. 5
  2. 4
  3. 20
  4. 10
Show Answer and Explanation

Correct answer: D. 10

Every unordered pair of vertices forms an edge: 5(5−1)/2.

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Practice Set 7 · Question 69 of 200

Question 69. How many edges does a simple cycle graph C5 have?

  1. 6
  2. 10
  3. 5
  4. 4
Show Answer and Explanation

Correct answer: C. 5

A cycle has one edge for each of its 5 vertices.

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Practice Set 7 · Question 70 of 200

Question 70. How many edges are in the path graph P5 with 5 vertices?

  1. 10
  2. 4
  3. 5
  4. 6
Show Answer and Explanation

Correct answer: B. 4

A path through 5 vertices uses 5−1 edges.

Continue to Practice Set 8

Practice Set 8 · Question 71 of 200

Question 71. How many edges are in the complete graph K6?

  1. 5
  2. 30
  3. 15
  4. 6
Show Answer and Explanation

Correct answer: C. 15

Every unordered pair of vertices forms an edge: 6(6−1)/2.

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Practice Set 8 · Question 72 of 200

Question 72. How many edges does a simple cycle graph C6 have?

  1. 15
  2. 6
  3. 5
  4. 7
Show Answer and Explanation

Correct answer: B. 6

A cycle has one edge for each of its 6 vertices.

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Practice Set 8 · Question 73 of 200

Question 73. How many edges are in the path graph P6 with 6 vertices?

  1. 5
  2. 6
  3. 7
  4. 12
Show Answer and Explanation

Correct answer: A. 5

A path through 6 vertices uses 6−1 edges.

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Practice Set 8 · Question 74 of 200

Question 74. How many edges are in the complete graph K7?

  1. 42
  2. 21
  3. 7
  4. 6
Show Answer and Explanation

Correct answer: B. 21

Every unordered pair of vertices forms an edge: 7(7−1)/2.

Next Question →

Practice Set 8 · Question 75 of 200

Question 75. How many edges does a simple cycle graph C7 have?

  1. 7
  2. 6
  3. 8
  4. 21
Show Answer and Explanation

Correct answer: A. 7

A cycle has one edge for each of its 7 vertices.

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Practice Set 8 · Question 76 of 200

Question 76. How many edges are in the path graph P7 with 7 vertices?

  1. 7
  2. 8
  3. 14
  4. 6
Show Answer and Explanation

Correct answer: D. 6

A path through 7 vertices uses 7−1 edges.

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Practice Set 8 · Question 77 of 200

Question 77. How many edges are in the complete graph K8?

  1. 28
  2. 8
  3. 7
  4. 56
Show Answer and Explanation

Correct answer: A. 28

Every unordered pair of vertices forms an edge: 8(8−1)/2.

Next Question →

Practice Set 8 · Question 78 of 200

Question 78. How many edges does a simple cycle graph C8 have?

  1. 7
  2. 9
  3. 28
  4. 8
Show Answer and Explanation

Correct answer: D. 8

A cycle has one edge for each of its 8 vertices.

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Practice Set 8 · Question 79 of 200

Question 79. How many edges are in the path graph P8 with 8 vertices?

  1. 9
  2. 16
  3. 7
  4. 8
Show Answer and Explanation

Correct answer: C. 7

A path through 8 vertices uses 8−1 edges.

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Practice Set 8 · Question 80 of 200

Question 80. How many edges are in the complete graph K9?

  1. 9
  2. 8
  3. 72
  4. 36
Show Answer and Explanation

Correct answer: D. 36

Every unordered pair of vertices forms an edge: 9(9−1)/2.

Continue to Practice Set 9

Practice Set 9 · Question 81 of 200

Question 81. How many edges does a simple cycle graph C9 have?

  1. 10
  2. 36
  3. 9
  4. 8
Show Answer and Explanation

Correct answer: C. 9

A cycle has one edge for each of its 9 vertices.

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Practice Set 9 · Question 82 of 200

Question 82. How many edges are in the path graph P9 with 9 vertices?

  1. 18
  2. 8
  3. 9
  4. 10
Show Answer and Explanation

Correct answer: B. 8

A path through 9 vertices uses 9−1 edges.

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Practice Set 9 · Question 83 of 200

Question 83. How many edges are in the complete graph K10?

  1. 9
  2. 90
  3. 45
  4. 10
Show Answer and Explanation

Correct answer: C. 45

Every unordered pair of vertices forms an edge: 10(10−1)/2.

Next Question →

Practice Set 9 · Question 84 of 200

Question 84. How many edges does a simple cycle graph C10 have?

  1. 45
  2. 10
  3. 9
  4. 11
Show Answer and Explanation

Correct answer: B. 10

A cycle has one edge for each of its 10 vertices.

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Practice Set 9 · Question 85 of 200

Question 85. How many edges are in the path graph P10 with 10 vertices?

  1. 9
  2. 10
  3. 11
  4. 20
Show Answer and Explanation

Correct answer: A. 9

A path through 10 vertices uses 10−1 edges.

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Practice Set 9 · Question 86 of 200

Question 86. How many edges are in the complete graph K11?

  1. 110
  2. 55
  3. 11
  4. 10
Show Answer and Explanation

Correct answer: B. 55

Every unordered pair of vertices forms an edge: 11(11−1)/2.

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Practice Set 9 · Question 87 of 200

Question 87. How many edges does a simple cycle graph C11 have?

  1. 11
  2. 10
  3. 12
  4. 55
Show Answer and Explanation

Correct answer: A. 11

A cycle has one edge for each of its 11 vertices.

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Practice Set 9 · Question 88 of 200

Question 88. How many edges are in the path graph P11 with 11 vertices?

  1. 11
  2. 12
  3. 22
  4. 10
Show Answer and Explanation

Correct answer: D. 10

A path through 11 vertices uses 11−1 edges.

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Practice Set 9 · Question 89 of 200

Question 89. A tree has 5 vertices. How many edges does it have?

  1. 8
  2. 4
  3. 5
  4. 6
Show Answer and Explanation

Correct answer: B. 4

Every finite tree has exactly |V|−1 edges.

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Practice Set 9 · Question 90 of 200

Question 90. How many leaves does the star graph K₁,4 have?

  1. 4
  2. 3
  3. 5
  4. 1
Show Answer and Explanation

Correct answer: A. 4

All 4 outer vertices have degree one.

Continue to Practice Set 10

Practice Set 10 · Question 91 of 200

Question 91. A tree has 6 vertices. How many edges does it have?

  1. 5
  2. 6
  3. 7
  4. 10
Show Answer and Explanation

Correct answer: A. 5

Every finite tree has exactly |V|−1 edges.

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Practice Set 10 · Question 92 of 200

Question 92. How many leaves does the star graph K₁,5 have?

  1. 4
  2. 6
  3. 1
  4. 5
Show Answer and Explanation

Correct answer: D. 5

All 5 outer vertices have degree one.

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Practice Set 10 · Question 93 of 200

Question 93. A tree has 7 vertices. How many edges does it have?

  1. 7
  2. 8
  3. 12
  4. 6
Show Answer and Explanation

Correct answer: D. 6

Every finite tree has exactly |V|−1 edges.

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Practice Set 10 · Question 94 of 200

Question 94. How many leaves does the star graph K₁,6 have?

  1. 7
  2. 1
  3. 6
  4. 5
Show Answer and Explanation

Correct answer: C. 6

All 6 outer vertices have degree one.

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Practice Set 10 · Question 95 of 200

Question 95. A tree has 8 vertices. How many edges does it have?

  1. 9
  2. 14
  3. 7
  4. 8
Show Answer and Explanation

Correct answer: C. 7

Every finite tree has exactly |V|−1 edges.

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Practice Set 10 · Question 96 of 200

Question 96. How many leaves does the star graph K₁,7 have?

  1. 1
  2. 7
  3. 6
  4. 8
Show Answer and Explanation

Correct answer: B. 7

All 7 outer vertices have degree one.

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Practice Set 10 · Question 97 of 200

Question 97. A tree has 9 vertices. How many edges does it have?

  1. 16
  2. 8
  3. 9
  4. 10
Show Answer and Explanation

Correct answer: B. 8

Every finite tree has exactly |V|−1 edges.

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Practice Set 10 · Question 98 of 200

Question 98. How many leaves does the star graph K₁,8 have?

  1. 8
  2. 7
  3. 9
  4. 1
Show Answer and Explanation

Correct answer: A. 8

All 8 outer vertices have degree one.

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Practice Set 10 · Question 99 of 200

Question 99. A tree has 10 vertices. How many edges does it have?

  1. 9
  2. 10
  3. 11
  4. 18
Show Answer and Explanation

Correct answer: A. 9

Every finite tree has exactly |V|−1 edges.

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Practice Set 10 · Question 100 of 200

Question 100. How many leaves does the star graph K₁,9 have?

  1. 8
  2. 10
  3. 1
  4. 9
Show Answer and Explanation

Correct answer: D. 9

All 9 outer vertices have degree one.

Continue to Practice Set 11

Practice Set 11 · Question 101 of 200

Question 101. A tree has 11 vertices. How many edges does it have?

  1. 11
  2. 12
  3. 20
  4. 10
Show Answer and Explanation

Correct answer: D. 10

Every finite tree has exactly |V|−1 edges.

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Practice Set 11 · Question 102 of 200

Question 102. How many leaves does the star graph K₁,10 have?

  1. 11
  2. 1
  3. 10
  4. 9
Show Answer and Explanation

Correct answer: C. 10

All 10 outer vertices have degree one.

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Practice Set 11 · Question 103 of 200

Question 103. A tree has 12 vertices. How many edges does it have?

  1. 13
  2. 22
  3. 11
  4. 12
Show Answer and Explanation

Correct answer: C. 11

Every finite tree has exactly |V|−1 edges.

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Practice Set 11 · Question 104 of 200

Question 104. How many leaves does the star graph K₁,11 have?

  1. 1
  2. 11
  3. 10
  4. 12
Show Answer and Explanation

Correct answer: B. 11

All 11 outer vertices have degree one.

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Practice Set 11 · Question 105 of 200

Question 105. In the complete graph K4, what is the degree of each vertex?

  1. 4
  2. 2
  3. 8
  4. 3
Show Answer and Explanation

Correct answer: D. 3

Each vertex is adjacent to the other 4−1 vertices.

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Practice Set 11 · Question 106 of 200

Question 106. An undirected graph has 6 edges. What is the sum of all vertex degrees?

  1. 13
  2. 36
  3. 12
  4. 6
Show Answer and Explanation

Correct answer: C. 12

By the handshaking lemma, the sum of degrees is 2|E|=12.

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Practice Set 11 · Question 107 of 200

Question 107. In the complete graph K5, what is the degree of each vertex?

  1. 3
  2. 10
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: C. 4

Each vertex is adjacent to the other 5−1 vertices.

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Practice Set 11 · Question 108 of 200

Question 108. An undirected graph has 7 edges. What is the sum of all vertex degrees?

  1. 49
  2. 14
  3. 7
  4. 15
Show Answer and Explanation

Correct answer: B. 14

By the handshaking lemma, the sum of degrees is 2|E|=14.

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Practice Set 11 · Question 109 of 200

Question 109. In the complete graph K6, what is the degree of each vertex?

  1. 12
  2. 5
  3. 6
  4. 4
Show Answer and Explanation

Correct answer: B. 5

Each vertex is adjacent to the other 6−1 vertices.

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Practice Set 11 · Question 110 of 200

Question 110. An undirected graph has 8 edges. What is the sum of all vertex degrees?

  1. 16
  2. 8
  3. 17
  4. 64
Show Answer and Explanation

Correct answer: A. 16

By the handshaking lemma, the sum of degrees is 2|E|=16.

Continue to Practice Set 12

Practice Set 12 · Question 111 of 200

Question 111. In the complete graph K7, what is the degree of each vertex?

  1. 6
  2. 7
  3. 5
  4. 14
Show Answer and Explanation

Correct answer: A. 6

Each vertex is adjacent to the other 7−1 vertices.

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Practice Set 12 · Question 112 of 200

Question 112. An undirected graph has 9 edges. What is the sum of all vertex degrees?

  1. 9
  2. 19
  3. 81
  4. 18
Show Answer and Explanation

Correct answer: D. 18

By the handshaking lemma, the sum of degrees is 2|E|=18.

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Practice Set 12 · Question 113 of 200

Question 113. In the complete graph K8, what is the degree of each vertex?

  1. 8
  2. 6
  3. 16
  4. 7
Show Answer and Explanation

Correct answer: D. 7

Each vertex is adjacent to the other 8−1 vertices.

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Practice Set 12 · Question 114 of 200

Question 114. An undirected graph has 10 edges. What is the sum of all vertex degrees?

  1. 21
  2. 100
  3. 20
  4. 10
Show Answer and Explanation

Correct answer: C. 20

By the handshaking lemma, the sum of degrees is 2|E|=20.

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Practice Set 12 · Question 115 of 200

Question 115. In the complete graph K9, what is the degree of each vertex?

  1. 7
  2. 18
  3. 8
  4. 9
Show Answer and Explanation

Correct answer: C. 8

Each vertex is adjacent to the other 9−1 vertices.

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Practice Set 12 · Question 116 of 200

Question 116. An undirected graph has 11 edges. What is the sum of all vertex degrees?

  1. 121
  2. 22
  3. 11
  4. 23
Show Answer and Explanation

Correct answer: B. 22

By the handshaking lemma, the sum of degrees is 2|E|=22.

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Practice Set 12 · Question 117 of 200

Question 117. In the complete graph K10, what is the degree of each vertex?

  1. 20
  2. 9
  3. 10
  4. 8
Show Answer and Explanation

Correct answer: B. 9

Each vertex is adjacent to the other 10−1 vertices.

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Practice Set 12 · Question 118 of 200

Question 118. An undirected graph has 12 edges. What is the sum of all vertex degrees?

  1. 24
  2. 12
  3. 25
  4. 144
Show Answer and Explanation

Correct answer: A. 24

By the handshaking lemma, the sum of degrees is 2|E|=24.

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Practice Set 12 · Question 119 of 200

Question 119. In the complete graph K11, what is the degree of each vertex?

  1. 10
  2. 11
  3. 9
  4. 22
Show Answer and Explanation

Correct answer: A. 10

Each vertex is adjacent to the other 11−1 vertices.

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Practice Set 12 · Question 120 of 200

Question 120. An undirected graph has 13 edges. What is the sum of all vertex degrees?

  1. 13
  2. 27
  3. 169
  4. 26
Show Answer and Explanation

Correct answer: D. 26

By the handshaking lemma, the sum of degrees is 2|E|=26.

Continue to Practice Set 13

Practice Set 13 · Question 121 of 200

Question 121. A connected graph with exactly two odd-degree vertices (case 1) has which property?

  1. It must be a complete graph.
  2. It has an Euler trail but no Euler circuit.
  3. It has an Euler circuit.
  4. It has no Euler trail.
Show Answer and Explanation

Correct answer: B. It has an Euler trail but no Euler circuit.

A connected graph has an Euler trail with different endpoints iff exactly two vertices have odd degree.

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Practice Set 13 · Question 122 of 200

Question 122. A connected graph has all vertices of even degree. What Eulerian feature does it have?

  1. An Euler circuit
  2. Exactly two odd-degree vertices
  3. No closed trail
  4. No edges
Show Answer and Explanation

Correct answer: A. An Euler circuit

A connected graph with all even vertex degrees has an Euler circuit.

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Practice Set 13 · Question 123 of 200

Question 123. How many odd-degree vertices may a finite undirected graph have?

  1. Any odd number
  2. Only one
  3. An even number
  4. Exactly three
Show Answer and Explanation

Correct answer: C. An even number

The handshaking lemma implies that the number of odd-degree vertices is even.

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Practice Set 13 · Question 124 of 200

Question 124. Which graph has an Euler circuit?

  1. A path with 4 vertices
  2. A cycle C5
  3. A star K₁,3
  4. A tree with 5 vertices
Show Answer and Explanation

Correct answer: B. A cycle C5

Every vertex of the cycle C5 has degree 2 and the graph is connected.

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Practice Set 13 · Question 125 of 200

Question 125. Does the path graph P5 have an Euler trail?

  1. Yes, but no Euler circuit
  2. Yes, with an Euler circuit
  3. No, it has four odd vertices
  4. Only if another edge is added
Show Answer and Explanation

Correct answer: A. Yes, but no Euler circuit

A nontrivial path has exactly two degree-one vertices and all other degrees equal 2.

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Practice Set 13 · Question 126 of 200

Question 126. Can a connected graph with four odd-degree vertices have an Euler trail that uses each edge once?

  1. Yes, always
  2. Only when it is complete
  3. No
  4. Only when it is planar
Show Answer and Explanation

Correct answer: C. No

An Euler trail in a connected graph requires exactly zero or two odd-degree vertices.

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Practice Set 13 · Question 127 of 200

Question 127. A simple cycle C6 has how many odd-degree vertices?

  1. 2
  2. 6
  3. 1
  4. 0
Show Answer and Explanation

Correct answer: D. 0

All six vertices have degree 2, so none has odd degree.

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Practice Set 13 · Question 128 of 200

Question 128. For a connected graph with exactly two odd vertices u and v, where must an Euler trail start and end?

  1. At the same vertex
  2. At u and v in either order
  3. At any two even vertices
  4. At vertices of highest degree
Show Answer and Explanation

Correct answer: B. At u and v in either order

In an open Euler trail the endpoints are exactly the odd-degree vertices.

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Practice Set 13 · Question 129 of 200

Question 129. Which property must be identical in two isomorphic simple graphs (case 1)?

  1. Their sorted vertex-degree sequences
  2. Their vertex labels
  3. Their drawing coordinates
  4. Their vertex names
Show Answer and Explanation

Correct answer: A. Their sorted vertex-degree sequences

Graph isomorphisms preserve adjacency and thus every vertex degree.

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Practice Set 13 · Question 130 of 200

Question 130. Two isomorphic simple graphs must have the same number of what?

  1. Vertex names
  2. Vertices and edges
  3. Drawing crossings
  4. Vertex labels
Show Answer and Explanation

Correct answer: B. Vertices and edges

An isomorphism is a bijection on vertices that preserves adjacency and thus the edge count.

Continue to Practice Set 14

Practice Set 14 · Question 131 of 200

Question 131. If one simple graph contains a triangle and another has no triangles, can they be isomorphic?

  1. Yes, always
  2. Only if both are trees
  3. No
  4. Only if their degrees match
Show Answer and Explanation

Correct answer: C. No

Graph isomorphisms preserve cycles of length three.

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Practice Set 14 · Question 132 of 200

Question 132. Can two graphs with different numbers of connected components be isomorphic?

  1. No
  2. Yes, if both are planar
  3. Yes, if edge counts match
  4. Yes, after relabelling
Show Answer and Explanation

Correct answer: A. No

An isomorphism preserves connectedness and the number of connected components.

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Practice Set 14 · Question 133 of 200

Question 133. Which pair of graphs is isomorphic?

  1. P4 and C4
  2. K3 and P3
  3. C4 and K4
  4. Two copies of P5 with different vertex labels
Show Answer and Explanation

Correct answer: D. Two copies of P5 with different vertex labels

Changing vertex labels does not alter the graph’s adjacency structure.

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Practice Set 14 · Question 134 of 200

Question 134. Is having the same sorted degree sequence sufficient to guarantee that two simple graphs are isomorphic?

  1. Yes, always
  2. No, not in general
  3. Only if both have an even number of vertices
  4. Only for directed graphs
Show Answer and Explanation

Correct answer: B. No, not in general

Nonisomorphic graphs can have identical sorted degree sequences.

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Practice Set 14 · Question 135 of 200

Question 135. If G is isomorphic to a tree T, which statement must hold?

  1. G is also a tree
  2. G must be complete
  3. G must have a cycle
  4. G has no vertices
Show Answer and Explanation

Correct answer: A. G is also a tree

Graph isomorphisms preserve connectedness and the absence of cycles.

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Practice Set 14 · Question 136 of 200

Question 136. If two graphs are isomorphic, what happens to vertex adjacency?

  1. It is always reversed
  2. It disappears
  3. It is preserved under a bijection
  4. It depends on drawing coordinates
Show Answer and Explanation

Correct answer: C. It is preserved under a bijection

The defining bijection of an isomorphism preserves adjacency.

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Practice Set 14 · Question 137 of 200

Question 137. A weighted graph has edges A–B of weight 2, B–C of weight 3, and A–C of weight 8. What is the shortest A-to-C distance?

  1. 8
  2. 6
  3. 13
  4. 5
Show Answer and Explanation

Correct answer: D. 5

Compare the direct route (8) with A–B–C (2+3=5); choose the smaller.

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Practice Set 14 · Question 138 of 200

Question 138. A weighted graph has edges A–B of weight 3, B–C of weight 4, and A–C of weight 9. What is the shortest A-to-C distance?

  1. 8
  2. 16
  3. 7
  4. 9
Show Answer and Explanation

Correct answer: C. 7

Compare the direct route (9) with A–B–C (3+4=7); choose the smaller.

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Practice Set 14 · Question 139 of 200

Question 139. A weighted graph has edges A–B of weight 4, B–C of weight 5, and A–C of weight 10. What is the shortest A-to-C distance?

  1. 11
  2. 9
  3. 10
  4. 19
Show Answer and Explanation

Correct answer: B. 9

Compare the direct route (10) with A–B–C (4+5=9); choose the smaller.

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Practice Set 14 · Question 140 of 200

Question 140. A weighted graph has edges A–B of weight 5, B–C of weight 6, and A–C of weight 11. What is the shortest A-to-C distance?

  1. 11
  2. 12
  3. 22
  4. 13
Show Answer and Explanation

Correct answer: A. 11

Compare the direct route (11) with A–B–C (5+6=11); choose the smaller.

Continue to Practice Set 15

Practice Set 15 · Question 141 of 200

Question 141. A weighted graph has edges A–B of weight 6, B–C of weight 7, and A–C of weight 12. What is the shortest A-to-C distance?

  1. 14
  2. 25
  3. 13
  4. 12
Show Answer and Explanation

Correct answer: D. 12

Compare the direct route (12) with A–B–C (6+7=13); choose the smaller.

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Practice Set 15 · Question 142 of 200

Question 142. A weighted graph has edges A–B of weight 7, B–C of weight 8, and A–C of weight 13. What is the shortest A-to-C distance?

  1. 28
  2. 14
  3. 13
  4. 16
Show Answer and Explanation

Correct answer: C. 13

Compare the direct route (13) with A–B–C (7+8=15); choose the smaller.

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Practice Set 15 · Question 143 of 200

Question 143. A weighted graph has edges A–B of weight 8, B–C of weight 9, and A–C of weight 14. What is the shortest A-to-C distance?

  1. 15
  2. 14
  3. 18
  4. 31
Show Answer and Explanation

Correct answer: B. 14

Compare the direct route (14) with A–B–C (8+9=17); choose the smaller.

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Practice Set 15 · Question 144 of 200

Question 144. A weighted graph has edges A–B of weight 9, B–C of weight 10, and A–C of weight 15. What is the shortest A-to-C distance?

  1. 15
  2. 20
  3. 34
  4. 16
Show Answer and Explanation

Correct answer: A. 15

Compare the direct route (15) with A–B–C (9+10=19); choose the smaller.

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Practice Set 15 · Question 145 of 200

Question 145. Find ⌊3.7⌋.

  1. 2
  2. 5
  3. 3
  4. 4
Show Answer and Explanation

Correct answer: C. 3

The floor is the greatest integer not exceeding 3.7.

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Practice Set 15 · Question 146 of 200

Question 146. Find ⌈3.2⌉.

  1. 2
  2. 4
  3. 3
  4. 5
Show Answer and Explanation

Correct answer: B. 4

The ceiling is the least integer not less than 3.2.

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Practice Set 15 · Question 147 of 200

Question 147. Find ⌊4.7⌋.

  1. 6
  2. 4
  3. 5
  4. 3
Show Answer and Explanation

Correct answer: B. 4

The floor is the greatest integer not exceeding 4.7.

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Practice Set 15 · Question 148 of 200

Question 148. Find ⌈4.2⌉.

  1. 5
  2. 4
  3. 6
  4. 3
Show Answer and Explanation

Correct answer: A. 5

The ceiling is the least integer not less than 4.2.

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Practice Set 15 · Question 149 of 200

Question 149. Find ⌊5.7⌋.

  1. 5
  2. 6
  3. 4
  4. 7
Show Answer and Explanation

Correct answer: A. 5

The floor is the greatest integer not exceeding 5.7.

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Practice Set 15 · Question 150 of 200

Question 150. Find ⌈5.2⌉.

  1. 5
  2. 7
  3. 4
  4. 6
Show Answer and Explanation

Correct answer: D. 6

The ceiling is the least integer not less than 5.2.

Continue to Practice Set 16

Practice Set 16 · Question 151 of 200

Question 151. Find ⌊6.7⌋.

  1. 7
  2. 5
  3. 8
  4. 6
Show Answer and Explanation

Correct answer: D. 6

The floor is the greatest integer not exceeding 6.7.

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Practice Set 16 · Question 152 of 200

Question 152. Find ⌈6.2⌉.

  1. 8
  2. 5
  3. 7
  4. 6
Show Answer and Explanation

Correct answer: C. 7

The ceiling is the least integer not less than 6.2.

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Practice Set 16 · Question 153 of 200

Question 153. Find ⌊7.7⌋.

  1. 6
  2. 9
  3. 7
  4. 8
Show Answer and Explanation

Correct answer: C. 7

The floor is the greatest integer not exceeding 7.7.

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Practice Set 16 · Question 154 of 200

Question 154. Find ⌈7.2⌉.

  1. 6
  2. 8
  3. 7
  4. 9
Show Answer and Explanation

Correct answer: B. 8

The ceiling is the least integer not less than 7.2.

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Practice Set 16 · Question 155 of 200

Question 155. Find ⌊8.7⌋.

  1. 10
  2. 8
  3. 9
  4. 7
Show Answer and Explanation

Correct answer: B. 8

The floor is the greatest integer not exceeding 8.7.

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Practice Set 16 · Question 156 of 200

Question 156. Find ⌈8.2⌉.

  1. 9
  2. 8
  3. 10
  4. 7
Show Answer and Explanation

Correct answer: A. 9

The ceiling is the least integer not less than 8.2.

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Practice Set 16 · Question 157 of 200

Question 157. Find ⌊9.7⌋.

  1. 9
  2. 10
  3. 8
  4. 11
Show Answer and Explanation

Correct answer: A. 9

The floor is the greatest integer not exceeding 9.7.

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Practice Set 16 · Question 158 of 200

Question 158. Find ⌈9.2⌉.

  1. 9
  2. 11
  3. 8
  4. 10
Show Answer and Explanation

Correct answer: D. 10

The ceiling is the least integer not less than 9.2.

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Practice Set 16 · Question 159 of 200

Question 159. Find ⌊10.7⌋.

  1. 11
  2. 9
  3. 12
  4. 10
Show Answer and Explanation

Correct answer: D. 10

The floor is the greatest integer not exceeding 10.7.

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Practice Set 16 · Question 160 of 200

Question 160. Find ⌈10.2⌉.

  1. 12
  2. 9
  3. 11
  4. 10
Show Answer and Explanation

Correct answer: C. 11

The ceiling is the least integer not less than 10.2.

Continue to Practice Set 17

Practice Set 17 · Question 161 of 200

Question 161. For f(n)=2n^2+5n+1, which is a tight asymptotic bound as n→∞?

  1. Θ(n^2)
  2. Θ(n^3)
  3. Θ(1)
  4. o(n^2)
Show Answer and Explanation

Correct answer: A. Θ(n^2)

A polynomial with positive leading coefficient grows at the rate of its highest power, n^2.

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Practice Set 17 · Question 162 of 200

Question 162. For positive integers n→∞, which statement is correct? (2 is fixed)

  1. n^3 = o(n^2)
  2. n^2 = Θ(1)
  3. n^3 = O(n^2)
  4. n^2 = o(n^3)
Show Answer and Explanation

Correct answer: D. n^2 = o(n^3)

n^2/n^3=1/n→0, so the little-o statement holds.

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Practice Set 17 · Question 163 of 200

Question 163. For f(n)=3n^3+5n+1, which is a tight asymptotic bound as n→∞?

  1. Θ(n^4)
  2. Θ(1)
  3. o(n^3)
  4. Θ(n^3)
Show Answer and Explanation

Correct answer: D. Θ(n^3)

A polynomial with positive leading coefficient grows at the rate of its highest power, n^3.

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Practice Set 17 · Question 164 of 200

Question 164. For positive integers n→∞, which statement is correct? (3 is fixed)

  1. n^3 = Θ(1)
  2. n^4 = O(n^3)
  3. n^3 = o(n^4)
  4. n^4 = o(n^3)
Show Answer and Explanation

Correct answer: C. n^3 = o(n^4)

n^3/n^4=1/n→0, so the little-o statement holds.

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Practice Set 17 · Question 165 of 200

Question 165. For f(n)=4n^4+5n+1, which is a tight asymptotic bound as n→∞?

  1. Θ(1)
  2. o(n^4)
  3. Θ(n^4)
  4. Θ(n^5)
Show Answer and Explanation

Correct answer: C. Θ(n^4)

A polynomial with positive leading coefficient grows at the rate of its highest power, n^4.

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Practice Set 17 · Question 166 of 200

Question 166. For positive integers n→∞, which statement is correct? (4 is fixed)

  1. n^5 = O(n^4)
  2. n^4 = o(n^5)
  3. n^5 = o(n^4)
  4. n^4 = Θ(1)
Show Answer and Explanation

Correct answer: B. n^4 = o(n^5)

n^4/n^5=1/n→0, so the little-o statement holds.

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Practice Set 17 · Question 167 of 200

Question 167. For f(n)=5n^5+5n+1, which is a tight asymptotic bound as n→∞?

  1. o(n^5)
  2. Θ(n^5)
  3. Θ(n^6)
  4. Θ(1)
Show Answer and Explanation

Correct answer: B. Θ(n^5)

A polynomial with positive leading coefficient grows at the rate of its highest power, n^5.

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Practice Set 17 · Question 168 of 200

Question 168. For positive integers n→∞, which statement is correct? (5 is fixed)

  1. n^5 = o(n^6)
  2. n^6 = o(n^5)
  3. n^5 = Θ(1)
  4. n^6 = O(n^5)
Show Answer and Explanation

Correct answer: A. n^5 = o(n^6)

n^5/n^6=1/n→0, so the little-o statement holds.

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Practice Set 17 · Question 169 of 200

Question 169. For f(n)=6n^6+5n+1, which is a tight asymptotic bound as n→∞?

  1. Θ(n^6)
  2. Θ(n^7)
  3. Θ(1)
  4. o(n^6)
Show Answer and Explanation

Correct answer: A. Θ(n^6)

A polynomial with positive leading coefficient grows at the rate of its highest power, n^6.

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Practice Set 17 · Question 170 of 200

Question 170. For positive integers n→∞, which statement is correct? (6 is fixed)

  1. n^7 = o(n^6)
  2. n^6 = Θ(1)
  3. n^7 = O(n^6)
  4. n^6 = o(n^7)
Show Answer and Explanation

Correct answer: D. n^6 = o(n^7)

n^6/n^7=1/n→0, so the little-o statement holds.

Continue to Practice Set 18

Practice Set 18 · Question 171 of 200

Question 171. For f(n)=7n^7+5n+1, which is a tight asymptotic bound as n→∞?

  1. Θ(n^8)
  2. Θ(1)
  3. o(n^7)
  4. Θ(n^7)
Show Answer and Explanation

Correct answer: D. Θ(n^7)

A polynomial with positive leading coefficient grows at the rate of its highest power, n^7.

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Practice Set 18 · Question 172 of 200

Question 172. For positive integers n→∞, which statement is correct? (7 is fixed)

  1. n^7 = Θ(1)
  2. n^8 = O(n^7)
  3. n^7 = o(n^8)
  4. n^8 = o(n^7)
Show Answer and Explanation

Correct answer: C. n^7 = o(n^8)

n^7/n^8=1/n→0, so the little-o statement holds.

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Practice Set 18 · Question 173 of 200

Question 173. For f(n)=8n^8+5n+1, which is a tight asymptotic bound as n→∞?

  1. Θ(1)
  2. o(n^8)
  3. Θ(n^8)
  4. Θ(n^9)
Show Answer and Explanation

Correct answer: C. Θ(n^8)

A polynomial with positive leading coefficient grows at the rate of its highest power, n^8.

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Practice Set 18 · Question 174 of 200

Question 174. For positive integers n→∞, which statement is correct? (8 is fixed)

  1. n^9 = O(n^8)
  2. n^8 = o(n^9)
  3. n^9 = o(n^8)
  4. n^8 = Θ(1)
Show Answer and Explanation

Correct answer: B. n^8 = o(n^9)

n^8/n^9=1/n→0, so the little-o statement holds.

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Practice Set 18 · Question 175 of 200

Question 175. For f(n)=9n^9+5n+1, which is a tight asymptotic bound as n→∞?

  1. o(n^9)
  2. Θ(n^9)
  3. Θ(n^10)
  4. Θ(1)
Show Answer and Explanation

Correct answer: B. Θ(n^9)

A polynomial with positive leading coefficient grows at the rate of its highest power, n^9.

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Practice Set 18 · Question 176 of 200

Question 176. For positive integers n→∞, which statement is correct? (9 is fixed)

  1. n^9 = o(n^10)
  2. n^10 = o(n^9)
  3. n^9 = Θ(1)
  4. n^10 = O(n^9)
Show Answer and Explanation

Correct answer: A. n^9 = o(n^10)

n^9/n^10=1/n→0, so the little-o statement holds.

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Practice Set 18 · Question 177 of 200

Question 177. On the real numbers, classify f(x)=2x+1.

  1. Constant
  2. Neither increasing nor decreasing
  3. Strictly increasing
  4. Strictly decreasing
Show Answer and Explanation

Correct answer: C. Strictly increasing

The slope 2 is positive, so f increases strictly.

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Practice Set 18 · Question 178 of 200

Question 178. On the real numbers, classify f(x)=3x+2.

  1. Neither increasing nor decreasing
  2. Strictly increasing
  3. Strictly decreasing
  4. Constant
Show Answer and Explanation

Correct answer: B. Strictly increasing

The slope 3 is positive, so f increases strictly.

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Practice Set 18 · Question 179 of 200

Question 179. On the real numbers, classify f(x)=4x+3.

  1. Strictly increasing
  2. Strictly decreasing
  3. Constant
  4. Neither increasing nor decreasing
Show Answer and Explanation

Correct answer: A. Strictly increasing

The slope 4 is positive, so f increases strictly.

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Practice Set 18 · Question 180 of 200

Question 180. On the real numbers, classify f(x)=5x+4.

  1. Strictly decreasing
  2. Constant
  3. Neither increasing nor decreasing
  4. Strictly increasing
Show Answer and Explanation

Correct answer: D. Strictly increasing

The slope 5 is positive, so f increases strictly.

Continue to Practice Set 19

Practice Set 19 · Question 181 of 200

Question 181. On the real numbers, classify f(x)=6x+5.

  1. Constant
  2. Neither increasing nor decreasing
  3. Strictly increasing
  4. Strictly decreasing
Show Answer and Explanation

Correct answer: C. Strictly increasing

The slope 6 is positive, so f increases strictly.

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Practice Set 19 · Question 182 of 200

Question 182. On the real numbers, classify f(x)=7x+6.

  1. Neither increasing nor decreasing
  2. Strictly increasing
  3. Strictly decreasing
  4. Constant
Show Answer and Explanation

Correct answer: B. Strictly increasing

The slope 7 is positive, so f increases strictly.

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Practice Set 19 · Question 183 of 200

Question 183. On the real numbers, classify f(x)=8x+7.

  1. Strictly increasing
  2. Strictly decreasing
  3. Constant
  4. Neither increasing nor decreasing
Show Answer and Explanation

Correct answer: A. Strictly increasing

The slope 8 is positive, so f increases strictly.

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Practice Set 19 · Question 184 of 200

Question 184. On the real numbers, classify f(x)=9x+8.

  1. Strictly decreasing
  2. Constant
  3. Neither increasing nor decreasing
  4. Strictly increasing
Show Answer and Explanation

Correct answer: D. Strictly increasing

The slope 9 is positive, so f increases strictly.

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Practice Set 19 · Question 185 of 200

Question 185. For f(x)=x², find f(2).

  1. 7
  2. 4
  3. 5
  4. 6
Show Answer and Explanation

Correct answer: B. 4

Squaring 2 gives 4.

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Practice Set 19 · Question 186 of 200

Question 186. For f(x)=x², find f(3).

  1. 9
  2. 6
  3. 10
  4. 5
Show Answer and Explanation

Correct answer: A. 9

Squaring 3 gives 9.

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Practice Set 19 · Question 187 of 200

Question 187. For f(x)=x², find f(4).

  1. 8
  2. 17
  3. 6
  4. 16
Show Answer and Explanation

Correct answer: D. 16

Squaring 4 gives 16.

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Practice Set 19 · Question 188 of 200

Question 188. For f(x)=x², find f(5).

  1. 26
  2. 7
  3. 25
  4. 10
Show Answer and Explanation

Correct answer: C. 25

Squaring 5 gives 25.

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Practice Set 19 · Question 189 of 200

Question 189. For f(x)=x², find f(6).

  1. 8
  2. 36
  3. 12
  4. 37
Show Answer and Explanation

Correct answer: B. 36

Squaring 6 gives 36.

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Practice Set 19 · Question 190 of 200

Question 190. For f(x)=x², find f(7).

  1. 49
  2. 14
  3. 50
  4. 9
Show Answer and Explanation

Correct answer: A. 49

Squaring 7 gives 49.

Continue to Practice Set 20

Practice Set 20 · Question 191 of 200

Question 191. For f(x)=x², find f(8).

  1. 16
  2. 65
  3. 10
  4. 64
Show Answer and Explanation

Correct answer: D. 64

Squaring 8 gives 64.

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Practice Set 20 · Question 192 of 200

Question 192. For f(x)=x², find f(9).

  1. 82
  2. 11
  3. 81
  4. 18
Show Answer and Explanation

Correct answer: C. 81

Squaring 9 gives 81.

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Practice Set 20 · Question 193 of 200

Question 193. Which statement about a connected undirected graph with 4 vertices is correct?

  1. There is a path between every pair of vertices.
  2. Every vertex has degree 3.
  3. It has no cycles.
  4. All edges have equal weights.
Show Answer and Explanation

Correct answer: A. There is a path between every pair of vertices.

This path condition is exactly the definition of connectedness.

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Practice Set 20 · Question 194 of 200

Question 194. A graph with exactly two connected components is what?

  1. Connected
  2. Disconnected
  3. Complete
  4. A tree
Show Answer and Explanation

Correct answer: B. Disconnected

A connected graph has exactly one connected component.

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Practice Set 20 · Question 195 of 200

Question 195. In an undirected graph, if there is a path from u to v, what must also exist?

  1. A path from v to u
  2. A loop at u
  3. An edge joining u and v directly
  4. An Euler circuit
Show Answer and Explanation

Correct answer: A. A path from v to u

An undirected path can be traversed in reverse.

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Practice Set 20 · Question 196 of 200

Question 196. How many connected components does a tree have?

  1. 0
  2. 2
  3. 1
  4. One per edge
Show Answer and Explanation

Correct answer: C. 1

A tree is connected by definition.

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Practice Set 20 · Question 197 of 200

Question 197. Which graph is necessarily connected?

  1. A graph with no edges on five vertices
  2. A disjoint union of two cycles
  3. A graph containing an isolated vertex and other vertices
  4. The complete graph K6
Show Answer and Explanation

Correct answer: D. The complete graph K6

Each pair of vertices of a complete graph is directly adjacent.

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Practice Set 20 · Question 198 of 200

Question 198. Can a connected graph with 6 vertices have only 4 edges?

  1. Yes, always
  2. No
  3. Only if it has loops
  4. Only if all degrees are even
Show Answer and Explanation

Correct answer: B. No

Any connected graph with n vertices has at least n−1 edges, here 5.

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Practice Set 20 · Question 199 of 200

Question 199. What is a bridge in a connected undirected graph?

  1. An edge whose removal disconnects the graph
  2. Any edge in a cycle
  3. A vertex of degree zero
  4. An edge joining a vertex to itself
Show Answer and Explanation

Correct answer: A. An edge whose removal disconnects the graph

A bridge is an edge whose deletion increases the number of connected components.

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Practice Set 20 · Question 200 of 200

Question 200. What is a spanning tree of a connected graph with 8 vertices?

  1. A cycle of length 8
  2. A subgraph with 8 edges
  3. A connected acyclic subgraph containing all 8 vertices
  4. Any disconnected forest
Show Answer and Explanation

Correct answer: C. A connected acyclic subgraph containing all 8 vertices

A spanning tree includes all vertices, is connected and acyclic, and has 7 edges.

Restart from Question 1

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